Here are my thoughts and my (not very substantial) guesses.

The pronunciation of *experiment* with the "merry" vowel (which is the same as the "square" vowel for speakers with the *merry-Mary* merger, and the same as the "nurse" vowel for speakers with the "[*ferry–furry* merger][1]") seems likely to be more widespread: as indicated in the original question, it's the main pronunciation given by dictionaries. It's also the pronunciation that would be "expected" based on theoretical considerations: a single vowel letter (other than <u>) in a stressed syllable followed by an unstressed, non-word-final syllable tends to correspond to a "short" vowel in pronunciation (this is sometimes called "Luick's Law"; it's also related to the phenomenon called "trisyllabic laxing", although that name is usually used to refer to some kind of process supposed to specifically affect  the pronunciation of certain kinds of derived words, and it seems a bit questionable to me to classify "experiment" as "derived" because even though we can identify an ending *-ment* and an initial element *experi-* that occur in other words, both would be bound morphemes; no word like *\*experi* occurs in English as a free base). 

I suspect the pronunciation of *experiment* with the "near" vowel arose either due to influence of the spelling (the pronunciation of "e" in contexts like this tends to be rather unpredictable), the influence of the pronunciation of the related word *experience* (where the "near" vowel is regular because of the [occurence of unstressed "i" before another vowel in the next syllable][2]), or some combination of both. 

The phonetic similarity of the vowels might also have contributed to the development and maintenance of the variation. (Some other words can be pronounced with either of these vowels, such as *feral, [query][3], inherent, coherent, adherent, hysteria*—although in these words the "near" vowel is actually preferred by prescriptivists because the vowel occurs in a stressed penultimate syllable, or before unstressed "i" followed by another vowel—and *(atmo)spheric(al),* for which most prescriptive sources seem to prefer the short vowel, but the long vowel of "near" seems to be common, probably in large part because of influence from the related noun *(atmo)sphere.*) 

I think spelling pronunciations and analogical changes tend to have less clearly defined regional distributions than regular sound changes/mergers, so I am not sure if it would be possible to determine any geographical trends. Like you, I haven't found any source that addresses this question.

The use of the "near" vowel is denounced by Charles Harrington Elster, author of the prescriptive pronunciation guide *The Big Book of Beastly Mispronunciations.* The relevant entry is available as part of the examples on PBS's website at "[What Speech Do We Like Best?][4]":

> **Experiment** ek-SPER-uh-mi̲nt or ek-SPAIR-uh-mi̲nt.  The first syllable is often, and acceptably, lightened to ik-.
>
Do not say ek-SPEER-uh-mi̲nt.  Properly, there is no *spear* in *experiment.* 


  [1]: https://en.wikipedia.org/wiki/English-language_vowel_changes_before_historic_/r/#Merry%E2%80%93Murray_merger
  [2]: https://english.stackexchange.com/questions/346984/why-is-salient-pronounced-with-a-long-a-sound
  [3]: https://english.stackexchange.com/questions/321830/kweh-ree-vs-kwee-ree-which-is-the-more-common-pronunciation-of-query/321906#321906
  [4]: http://www.pbs.org/speak/speech/beastly/#pagetop